Resolve `(x^(2)+5)/((x+2)(x+1))` into partial fractions

Answer

To resolve the expression \(\frac{x^2 + 5}{(x + 2)(x + 1)}\) into partial fractions, we will follow these steps:

### Step 1: Set up the partial fraction decomposition
We want to express the given fraction as a sum of simpler fractions. We can write:
\[ \frac{x^2 + 5}{(x + 2)(x + 1)} = \frac{A}{x + 2} + \frac{B}{x + 1} \]
where \(A\) and \(B\) are constants that we need to determine.

### Step 2: Clear the denominators
Multiply both sides by \((x + 2)(x + 1)\) to eliminate the denominators:
\[ x^2 + 5 = A(x + 1) + B(x + 2) \]

### Step 3: Expand the right-hand side
Now, expand the right-hand side:
\[ x^2 + 5 = Ax + A + Bx + 2B \]
Combine like terms:
\[ x^2 + 5 = (A + B)x + (A + 2B) \]

### Step 4: Set up equations based on coefficients
Now, we can equate the coefficients of \(x\) and the constant terms from both sides:
1. For \(x\): \(A + B = 0\) (Equation 1)
2. For the constant term: \(A + 2B = 5\) (Equation 2)

### Step 5: Solve the system of equations
From Equation 1, we can express \(A\) in terms of \(B\):
\[ A = -B \]
Substituting \(A = -B\) into Equation 2:
\[ (-B) + 2B = 5 \]
This simplifies to:
\[ B = 5 \]
Now substitute \(B = 5\) back into Equation 1 to find \(A\):
\[ A + 5 = 0 \implies A = -5 \]

### Step 6: Write the partial fraction decomposition
Now that we have \(A\) and \(B\), we can write the partial fraction decomposition:
\[ \frac{x^2 + 5}{(x + 2)(x + 1)} = \frac{-5}{x + 2} + \frac{5}{x + 1} \]

### Final Answer
Thus, the partial fraction decomposition of \(\frac{x^2 + 5}{(x + 2)(x + 1)}\) is:
\[ \frac{-5}{x + 2} + \frac{5}{x + 1} \]

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